On local non-zero constraints in PDE with analytic coefficients

Abstract

We consider the Helmholtz equation with real analytic coefficients on a bounded domain $\Omega \subset {\mathbb{R}}^d$. We take $d+1$ prescribed boundary conditions $f^i$ and frequencies $\omega$ in a fixed interval $[a,b]$. We consider a constraint on the solutions $u_{\omega }^i$ of the form $\zeta (u_{\omega }^1,\dots ,u_{\omega }^{d+1},\nabla u_{\omega }^1,\dots ,\nabla u_{\omega }^{d+1})\ne 0$, where $\zeta$ is analytic, which is satisfied in $\Omega$ when $\omega =0$. We show that for any ${\Omega }^{\prime }⋐\Omega$ and almost any $d+1$ frequencies ${\omega }_k$ in $[a,b]$, there exist $d+1$ subdomains ${\Omega }_k$ such that ${\Omega }^{\prime }\subset {\cup }_k{\Omega }_k$ and $\zeta (u_{{\omega }_k}^1,\dots ,u_{{\omega }_k}^{d+1},\nabla u_{{\omega }_k}^1,\dots ,\nabla u_{{\omega }_k}^{d+1})\ne 0$ in ${\Omega }_k$. This question comes from hybrid imaging inverse problems. The method used is not specific to the Helmholtz model and can be applied to other frequency dependent problems.

9 pages, 1 figure

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